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If f(x)="cos"(tan^(-1)x), then the value...

If `f(x)="cos"(tan^(-1)x),` then the value of the integral `int_0^1xf''(x)dx" is"` (a) `(3-sqrt(2))/2` (b) `(3+sqrt(2))/2` (c) `1` (d) `1-3/(2sqrt(2))`

A

`(3-sqrt(2))/2`

B

`(3+sqrt(2))/2`

C

`1`

D

`1-3/(2sqrt(2))`

Text Solution

Verified by Experts

The correct Answer is:
D
`f(x)=cos(tan^(-1)x)`
`:.f'(x)=-(sin (tan^(-1)x))/(1+x^(2))`
`:. I=int_(0)^(1)x f''(x)dx`
`=[f'(x)]_(0)^(1)-int_(0)^(1)f'(x)dx` (Integrating by parts)
`=[f'(1)]-[f(x)]_(0)^(1)`
`=f'(1)-f(1)+f(0)`
Now `f(0)=1,f'(1)=-1/(2sqrt(2)),f(1)=1/(sqrt(2))`
`:. I=1-3/(2sqrt(2))`
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